Average Error: 15.3 → 1.4
Time: 8.6s
Precision: binary64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\cos M}\\ \left(t_0 \cdot {t_0}^{2}\right) \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)} \end{array} \]
(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))
↓
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cbrt (cos M))))
   (*
    (* t_0 (pow t_0 2.0))
    (exp (- (fabs (- m n)) (+ (pow (- (/ (+ m n) 2.0) M) 2.0) l))))))
double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}
↓
double code(double K, double m, double n, double M, double l) {
	double t_0 = cbrt(cos(M));
	return (t_0 * pow(t_0, 2.0)) * exp((fabs((m - n)) - (pow((((m + n) / 2.0) - M), 2.0) + l)));
}

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}
↓
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cbrt(Math.cos(M));
	return (t_0 * Math.pow(t_0, 2.0)) * Math.exp((Math.abs((m - n)) - (Math.pow((((m + n) / 2.0) - M), 2.0) + l)));
}

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end
↓
function code(K, m, n, M, l)
	t_0 = cbrt(cos(M))
	return Float64(Float64(t_0 * (t_0 ^ 2.0)) * exp(Float64(abs(Float64(m - n)) - Float64((Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0) + l))))
end

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Power[N[Cos[M], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\cos M}\\
\left(t_0 \cdot {t_0}^{2}\right) \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)}
\end{array}

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

    \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

  2. Simplified15.3

    \[\leadsto \color{blue}{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)}} \]

  3. Taylor expanded in K around 0 1.4

    \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)} \]

  4. Simplified1.4

    \[\leadsto \color{blue}{\cos M} \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)} \]

  5. Applied egg-rr1.4

    \[\leadsto \color{blue}{\left(\sqrt[3]{\cos M} \cdot {\left(\sqrt[3]{\cos M}\right)}^{2}\right)} \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)} \]

  6. Final simplification1.4

    \[\leadsto \left(\sqrt[3]{\cos M} \cdot {\left(\sqrt[3]{\cos M}\right)}^{2}\right) \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)} \]

Reproduce

herbie shell --seed 2022141 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))